Validity of the acoustic approximation for elasticwaves in heterogeneous media

Philippe Cance1 and Yann Capdeville1

ABSTRACT

The acoustic approximation of elastic waves is a verycommon approximation in exploration geophysics. The in-terest of the acoustic approximation in the inverse problemcontext lies in the fact that it leads to a much lower numericalcost than for the elastic problem. Nevertheless, the earth isnot an acoustic body, and it has been found in the past thatthis approximation is not without drawbacks mainly becauseof P-to-S energy conversion and that anisotropy cannot beeasily modeled. We studied a different issue of this approxi-mation related to small heterogeneities with respect to theminimum wavelength of the wavefield. We first numericallyfound that elastic and acoustic waves behave differentlywith respect to small-scale heterogeneities, introducingdifferences in amplitudes but also in phase between elasticand acoustic signals. We then derived physical and math-ematical interpretations of this phenomenon, developingthe different nature of elastic- and acoustic-wave propaga-tion that led to the conclusion that, in rough media, acousticwaves can only be a poor-quality approximation of elasticwaves. Interestingly, we also found that, in the acoustic case,small-scale heterogeneities give rise to natural acoustic ef-fective anisotropic media through an anisotropic effectivemass matrix. Unfortunately, this anisotropy is of a differentnature compared with the elastic effective anisotropy andcan not be used to mimic elastic anisotropy.

INTRODUCTION

In the exploration geophysics context, the acoustic-wave equa-tion is often used as an approximation to the elastic-wave equation.This approximation is widely used for migration techniques (e.g.,

Berkhout, 1984; Etgen et al., 2009), full-waveform inversion tech-niques (Virieux and Operto, 2009), theoretical developments (e.g.,Tarantola, 1984; Gauthier et al., 1986; Bunks et al., 1995; Pratt et al.,1998; Ben-Hadj-Ali et al., 2008; Métivier et al., 2013), and dataapplications (e.g., Shin and Cha, 2008; Plessix et al., 2010; Vighet al., 2010; Schiemenz and Igel, 2013). The main reason to performsuch an approximation is to simplify forward and inverse problems.For forward modeling, the acoustic approximation considerably re-duces the numerical cost for two main reasons: First, acoustic equa-tions are scalar equations, whereas elastic equations rely on vectorand tensor quantities. Second, acoustic forward modeling may gainup to a factor of 100 in computation time because the space sam-pling (and then time sampling) of the minimum wavelength of thewavefield relies on P-wave velocities in the acoustic case, whereas itrelies on the much smaller S-wave velocities in the elastic case. Forthe full-waveform inverse problem, faster forward modeling is adecisive advantage, and having to invert for fewer parameters thanfor the elastic case leads to a simpler inverse problem. If the reasonsthat motivate such an approximation are clear, its justifications arenot that obvious. Indeed, the earth is not an acoustic body and, evenfor marine exploration setup for which sources and receivers lie inan acoustic medium, waves under consideration are elastic wavesfor most of the propagation path between the source and receiver. Inthe migration context, in which it is attempted to migrate only Parrivals, this approximation seems often justified, at least whenno anisotropy is present. In the full-waveform inversion context,however, this approximation does not always seem justified becauseit leads to persistent unreliable inversion results when S-waveparameters play a nonnegligible role in P-wave propagation (Barnesand Charara, 2008; Marelli et al., 2012). In the forward-modelingcontext, acoustic approximations of elastic P-waves are well-knownfor encountering major difficulties (Alkhalifah, 1998, 2000, 2003;Grechka et al., 2004; Fletcher et al., 2008; Operto et al., 2009; Bak-ker and Duveneck, 2011; Chu et al., 2012; Bube et al., 2012; Wuand Alkhalifah, 2014). Indeed, if the acoustic approximation ofelastic P-waves is natural for homogeneous isotropic media, it is

Manuscript received by the Editor 21 August 2014; revised manuscript received 22 December 2014; published online 10 June 2015.1Université de Nantes, Laboratoire de Planétologie et Géodynamique de Nantes, CNRS, Nantes, France. E-mail: philippe.cance@univ-nantes.fr;

yann.capdeville@univ-nantes.fr.© 2015 Society of Exploration Geophysicists. All rights reserved.

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GEOPHYSICS, VOL. 80, NO. 4 (JULY-AUGUST 2015); P. T161–T173, 8 FIGS., 1 TABLE.10.1190/GEO2014-0397.1

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not the case anymore for anisotropic media nor for heterogeneousmedia, in which P-to-S conversions may occur but cannot be takeninto account by acoustic equations (in some cases, one can find amethod for correcting this issue, e.g., in Hobro et al., 2014). Be-cause scattering and effective anisotropy are observed on elasticwavefields propagating with wavelengths much larger than the sizeof the medium’s heterogeneities, we focus here on the accuracy ofacoustic approximations for elastic P-waves in “rough” hetero-geneous media, that is, when heterogeneities with scales smallerthan the minimum wavelength are present. To address this problem,we first recall the elastic- and acoustic-wave equations and how theycan be related to each other in the homogeneous media case. Wethen present three simple numerical experiments to challenge theacoustic approximation of elastic-wave propagation. After showingthat the approximation fails for rough media, we mathematicallyand physically propose an explanation for this disagreement basedon homogenization theory. Finally, we discuss the induced effectiveacoustic anisotropy versus elastic anisotropy in the simple layeredcase before concluding.

ACOUSTIC APPROXIMATION OF THE ELASTICEQUATIONS

Considering an elastic domain Ω with a free surface ∂Ω, the elas-tic-wave equations driving the displacement vector uðx; tÞ are

ρu − ∇ · σ ¼ f in Ω; (1)

σ ¼ c∶ϵðuÞ in Ω; (2)

σ · n ¼ 0 in ∂Ω; (3)

where x is the positional vector, t is the time component, cðxÞ is theelastic stiffness tensor, ρðxÞ is the (mass) density, σ is the stresstensor, ϵ ¼ ð∇uþ T∇uÞ∕2 is the strain tensor, T denotes the trans-pose operator, f is the source term, and n denotes the unit normal to∂Ω. In general, the elastic stiffness tensor c depends upon 21 inde-pendent coefficients in 3D and six in 2D, but only on the two Laméparameters λðxÞ and μðxÞ for the isotropic case (Lamé, 1852). Insuch a case, the constitutive relation 2 can be rewritten as

σ ¼ λtrðϵÞIþ 2μϵ; (4)

where I is the identity tensor and trðAÞ ¼ Aii (using the Einsteinimplicit summation convention) is the trace operator. Isotropic P-and S-wave velocities are then related to Lamé parameters through

λþ 2μ ¼ ρVP2; (5)

μ ¼ ρVS2: (6)

For the same domain Ω, but for the acoustic-wave propagation,the velocity potential q is the solution of

1

κq − ∇ · u ¼ g in Ω; (7)

u ¼ 1

ρ∇q in Ω; (8)

q ¼ 0 in ∂Ω; (9)

where κðxÞ is the acoustic bulk modulus, ρðxÞ is the (mass) density,uðx; tÞ is the displacement vector, and gðx; tÞ is a scalar source term.In general, the acoustic medium is fully described by only twoparameters, namely, its density ρðxÞ and the “sound speed”VðxÞ, such that κðxÞ ¼ ρðxÞV2ðxÞ, where the acoustic bulk modulusκ shall not be confused with the elastic bulk modulus K ¼ λþ 2

3μ

(which is not used in this paper).In general, the displacement vector u solution to the above elas-

todynamic equations can be related to its acoustic counterpart onlyfor an infinite homogeneous isotropic domain and for an explosiveisotropic source as described in Appendix A. However, realisticseismic wave propagation in the earth necessitates at least a freesurface and some reflectors. Reflections will then occur at each in-terface generating S waves, which breaks the assumption that thedisplacement u can derive only from the above potential q. A sol-ution to this problem is to “filter” rotational waves (S waves), keep-ing only the irrotational part of the signal, which ought to derivefrom the wanted potential. To do so, we use the pressure wavefieldp defined as p ¼ −ð1∕2Þκ∇ · u in the elastic case (Landau and Lif-shitz, 1959b). Indeed, no rotational part of the original displacementu is left in p because ∇ · ð∇ ×mÞ ¼ 0 for any vector m. We thencompare the elastic pressure p with the acoustic pressure signal de-fined as q (Landau and Lifshitz, 1959a). In practice, P-to-S (and S-to-P) conversions cannot be accounted for by acoustic approxima-tions anyway, leading to incorrect modeling of amplitudes for elas-tic P-waves. Phases, however, may still be correctly modeled inisotropic homogeneous media (with reflections), at least for the firstarrivals. In the next two sections, we numerically investigate theequivalence of acoustic and elastic waves in nonhomogeneous me-dia with respect to the pressure signal’s phase.

Heterogeneous media: The smooth and rough cases

As shown in the previous section, the acoustic- and elastic-waveequation solution equivalence for P-waves only holds theoreticallyin the case of a constant infinite medium. Nevertheless, it isreasonable to assume that the approximation is still valid in thequasi-homogeneous (very smooth) media case. In practice, thisapproximation is used in heterogeneous media that most of the timecontain small-scale heterogeneities, media that we will refer to asrough media. In the next section, we numerically investigate thevalidity and the behavior of the acoustic approximation of elasticwaves in smooth and rough media.Before moving further, we define more precisely smooth and

rough media. To do so, we first define the medium roughness scaleλ0 (contrary to the Lamé parameter λ, wavelengths or scales, such asλ0 will always be subscripted). The scale λ0 is a characteristic meas-urement of the spatial variations of elastic- and acoustic-mediumparameters. For example, it can be related to the minimum thicknessof the layers in a discontinuous layered medium or to the smallestoscillation scale of a continuous medium. Second, we need an es-timate of the minimum wavelength of the wavefield λmin. Knowingthe maximum frequency fc (the corner frequency of the source ormaximum frequency of the filtered seismograms) and the minimumwave velocity over the whole domain Vmin, a lower bound estimateof λmin is Vmin∕fc. In general, S-waves being always slower than P-waves, Vmin should be set using the slowest S-waves. However, weare primarily interested in P-waves and we will use the minimum

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P-wave velocity to estimate Vmin. Defining the parameter ε0 ¼λ0∕λmin to describe the roughness of the medium toward the wave-field, the following two cases will be considered:

1) ε0 ≫ 1, the smooth medium case2) ε0 ≪ 1, the rough medium case.

Numerical experiments of the acoustic approximationfor heterogeneous smooth and rough media

To study the effect of the heterogeneity scale on the acousticapproximation of elastic P-waves, we use three different 2D experi-ments and for each experiment type, smooth and rough models areintroduced:

1) experiment 1: periodic continuous stratified media (Figure 1)2) experiment 2: 2D random media (Figure 2)3) experiment 3: 2D SEG advanced modeling (SEAM)-based

(Fehler and Larner, 2008) media (Figure 3).

For each medium, we first define the elastic version by definingthe density ρðxÞ, and either Lamé parameters λðxÞ and μðxÞ orP- and S-wave velocities VPðxÞ and VSðxÞ, respectively, over thephysical domain Ω. Then, the acoustic model version is definedas for the homogeneous case by density and P-wave velocitiesfrom the elastic case: V ¼ VP (and we therefore always haveκ ¼ λþ 2μ ¼ c1111 ¼ c2222).

Experiment 1 media

For experiment 1, the layered medium is continuously definedvertically (x2 component) as a sine function around an averagevalue for the density ρ and the Lamé parameters (see Figure 1):

ρðx2Þ ¼ hρi�1þ νρ cos

2πx2λ0

�; (10)

λðx2Þ ¼ hλi�1þ νλ cos

2πx2λ0

�; (11)

μðx2Þ ¼ hμi�1þ νμ cos

2πx2λ0

�; (12)

where λ0 is the period; hρi, hλi, and hμi are the average quantities;and νρ ¼ 20% and νλ ¼ νμ ¼ 55% are the contrasts on density andelastic parameters, respectively. These contrasts may not reflectrealistic media values, but they are chosen to make our point clear,as it will appear later. Knowing that using a source corner frequencyfc ¼ 10 Hz leads to λPmin

¼ 300 m and λSmin¼ 200 m, we choose

λ0 ¼ 8.5 km for the smooth medium (ε0 ∼ 28) and λ0 ¼ 50 m forthe rough medium (ε0 ∼ 0.17).

Experiment 2 media

For experiment 2, we first define the rough medium. It is definedas a matrix of 800 × 900 square elements of 20 × 20 m2 with con-stant elastic properties. For each element, the density and P- and S-wave velocities are randomly chosen within a contrast ν ¼ 12:5%of their average value (we can then consider that the Lamé coeffi-cients — or elastic coefficients — are roughly taken within a

contrast νc ≃ 35% of their average value). For a source corner fre-quency fc ¼ 10 Hz, we roughly have λPmin

≃ 200 m (ε0 ¼ 0.1) andλSmin

≃ 100 m, which is indeed large compared with λ0 ¼ 20 m.The smooth model is obtained by spatially low-pass filtering therough model’s density and velocities. To do so, we introduce(see Appendix B) a low-pass-filter operator F λcð:Þ such that, forany spatially varying quantity AðxÞ, F λcðAÞðxÞ does not contain

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Figure 1. Stratified media for experiment 1. (a and b) Density mapsof the rough and smooth media, respectively. In the rough case, wesaturate image contrasts for visibility and the lower left box is amagnification of the small-scale varying medium correspondingto the 1-km × 200-m central black box. The black star indicatesthe source location, and the white diamonds (white line) indicatethe receiver locations. (c) Vertical cross section for x ¼ 15 km ob-tained for λ0 ¼ 50 m (rough model, gray line) and λ0 ¼ 8.5 km(smooth model, black line) with a zoom (the lower left box corre-sponds to depths between 17 and 17.2 km).

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spatial variations smaller than λc. We use λ0 ¼ λc ¼ 1200 m to de-fine the smooth model (ε0 ¼ 6).

Experiment 3 media

For experiment 3, we use a 2D cross section of the SEAM (Fehlerand Larner, 2008), defined on a 10 × 10‐m regular grid. The roughmodel is obtained by applying a taper toward constant properties to

the sides of the model, and S-wave velocities under 1 km∕s havebeen clipped (implying that the top water layer has been replaced byan elastic medium). Small-scale heterogeneities (with a pseudoper-iod of λ0 ≃ 70 m) and strong contrasts ν on density, P-wave velocity(Figure 3c), and S-wave velocity can be found on top (ν ≃ 5%) andunder (ν ≃ 8%) the center salt structure. For a source corner fre-quency of fc ¼ 4 Hz, the estimated P-wave minimum wavelengthis λPmin

≃ 375 m, which can be considered to be much larger thanthe medium roughness scale λ0 ≃ 70 m (ε0 ∼ 0.2). Similar to ex-periment 2, the smooth model is obtained from the rough modelby applying the low-pass filter to the rough density and wave veloc-ities with λ0 ¼ 2500 m. Using the previous source corner frequencyfc ¼ 4 Hz then gives an estimated P-wave minimum wavelength ofλPmin

≃ 280 m, which is indeed much smaller than λ0. However, de-spite the important ratio ε0 ¼ λ0∕λPmin

≃ 9, important large-scalecontrasts on velocities over the domain do not allow for λPmin

tobe a representative length to ensure that the filtered medium is reallysmooth. Indeed, the medium has to be smooth with respect to themain wavelength of the wavefield, which is more accurately de-scribed by the central frequency of the source (f0 ≃ fc∕2.5)and for P-wave velocities encountered in the most part of the wave-path. We estimate this main P-wave velocity at minimum asVP ≃ 2500 m∕s, giving then a main wavelength of VP∕f0 ≃1600 m, which is not small enough compared with λ0. We cannotsmooth the medium any further (a correct cut-off frequency used toobtain a really smooth medium would then correspond to a wave-length of the order of the medium’s depth, thus rendering thesmooth medium almost homogeneous), and the only option leftto define a smooth medium using the more reasonable mediumroughness scale of λ0 ¼ 2500 m is to increase the source cornerfrequency to fc ¼ 30 Hz, allowing an estimated main wavelengthfor the wavefield of approximately 210 m ≪ λ0 (ε0 ∼ 12).

Numerical experiments setup

Each experiment is devised as a seismic reflection survey: Foreach experiment, the source and the receivers are located nearthe top surface. To simplify the wavefield analysis, the top boundaryis treated with an absorbing boundary condition instead of a free-surface boundary condition, thus eliminating the free-surface re-flected downgoing wave. Lateral domain boundaries are also de-fined as absorbing boundaries. The bottom boundary is set to afree-surface condition, which is unusual, but it remains a simpleway to generate an upcoming reflected wave. We then observe asimple direct P-wave (traveling only through a homogeneousmedium) and waves coming back to the surface after travelingthrough the heterogeneous medium. To numerically solve the elas-tic- and acoustic-wave equations, we use the spectral elementmethod (SEM) (Komatitsch and Vilotte, 1998; Komatitsch andTromp, 2002). The absorbing conditions are achieved using per-fectly matched layers (PMLs) (Festa and Vilotte’s [2005] version).The actual computation domains are made larger than the depictedone to avoid any spurious reflection from the PMLs. For each ex-periment, the same mesh, spectral element polynomial basis, andtime sampling are used in the elastic and acoustic cases. We sys-tematically mesh the discontinuities of the medium, if any, and el-ements size and number of 1D Gauss-Lobatto-Legendre (GLL)points are always chosen to oversample the wavefield (see Table 1).This allows us to obtain excellent numerical accuracy (see Appen-

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Figure 2. Random media for experiment 2. (a and b) Density mapsof the rough and smooth media, respectively. The rough model con-tains square elements with constant elastic properties of sizeλ0 ¼ 20 m. The smooth model is a low-pass-filtered version of therough model with a cut-off spatial frequency of λ0 ¼ 1200 m. Theblack star indicates the source location, and the white diamonds(white line) indicate the receiver locations. (c) Vertical cross sectionfor x ¼ 5 km obtained for λ0 ¼ 20 m (rough model, gray line) andλ0 ¼ 1200 m (smooth model, black line).

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dix C), so that the observed differences between signals cannot berelated to numerical artifacts.To assess the accuracy of the acoustic approximation of elastic

waves, we compare the acoustic pressure q with the elastic pressure−ð1∕2Þκ∇ · u, but not directly. Indeed, due to energy losses duringP-to-S conversions for elastic-wave propagation, the amplitude ofthe acoustic pressure is not expected to match the amplitude of theelastic pressure, and only the phase of the signal may be accuratelymodeled. For experiments 1 and 2, the signals are very simple, mak-ing it possible to use a simple and crude way to perform phase com-parisons: Each wave arrival is normalized to the same amplitude in agiven time window. Doing so, observed signal differences are onlyrelated to phase error.

Smooth media results

For the smooth case, results for experiments 1–3 are, respectively,gathered in the left columns of Figures 4–6. For experiments 1 and2, traces show mainly two well-separated arrivals — first the directwave and then the wave reflected at the domain bottom — whichare then individually normalized. For experiment 3, only one arrivalcan clearly be identified (Figure 6e) for almost every offset. Con-sequently, the normalization is applied on the whole signal and notby arrival time windows as it should be. The agreement betweenacoustic and elastic solutions for experiments 1 and 2 is excellent,and the relative error jðpel − pacÞ∕pelj between the normalized elas-tic pressure pel ¼ ∇ · u∕maxð∇ · uÞ and its (also normalized)acoustic counterpart pac ¼ q∕maxðqÞ is less than 1% (Figures 4eand 5e). For experiment 3, the agreement between the two solutionsis also very good (Figure 6e) except for some late arrivals in a smallarea. These late differences are due to P-to-S energy losses in thehigh-velocity structure area (the salt area).For smooth models, it can be concluded that, as expected, the

acoustic approximation of elastic waves is accurate, at least for timearrivals.

Rough media results

For the rough case, results for experiments 1–3 are, respectively,gathered in the right columns of Figures 4–6. For experiments 1 and2, the acoustic approximation is accurate for the direct wave arrival,which is expected because the wave propagation path is homo-geneous (Figures 4f and 5f). For the bottom reflected arrival, a sig-nificant phase shift is observed for experiments 1 and 2. This phaseshift is larger for increasing offsets for experiment 1 (Figure 4f),whereas it is almost offset independent for experiment 2 (Figure 5f).For experiment 3, observations are complicated by the fact that theamplitude normalization cannot be performed efficiently, but a sim-ilar phase shift as for experiments 1 and 2, even though smaller, canbe observed (Figure 6b and 6d).From these numerical observations, we conclude that something

more than just P-to-S energy conversion losses occurs during elas-tic-wave propagation in rough media: Ballistic acoustic and elasticwaves experience an overall different wavespeed even though thelocal wavespeed is by construction the same. This is an indicationthat the physics of acoustic- and elastic-wave propagation is differ-ent. To understand this observation better physically, we propose torely on the two-scale homogenization theory for elastic waves(Backus, 1962; Capdeville et al., 2010a, 2010b) and acoustic waves(treated as a special case of Guillot et al. [2010]). This is the purposeof the next section.

HOMOGENIZATION FOR THE WAVEPROPAGATION PROBLEM

In general, the two-scale homogenization method allows us tocompute large-scale-only “effective” parameters and equationsequivalent to an initial problem with large and fine scales. It allowsus to first better understand physical phenomena, in which smallscales have only an effective role and, second, compute solutionsof the initial problem at the effective scale only, thus generallysignificantly reducing the computing time and/or the meshing

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problem. Two-scale homogenization techniques have been studiedfor a long time for various problems in mechanics for periodic ornonperiodic media (Auriault and Sanchez-Palencia, 1977; Bensous-san et al., 1978; Sanchez-Palencia, 1980; Allaire, 1992).For wave propagation problems, two-scale homogenization al-

lows us to understand how small-scale heterogeneities are upscaled— or “seen” — by a long-wavelength wavefield, and therefore itgives us the opportunity to explain and interpret observations whenthe ratio of heterogeneity scale versus minimum wavelength is verysmall compared with 1∶ε0 ¼ λ0∕λmin ≪ 1. We first briefly intro-duce the homogenization procedures for elastic and acoustic wavesin general heterogeneous media. Then, we detail the specific caseof layered media — especially the periodic case — because they

present some further theoretical results allowing for easier interpre-tations. Indeed, Backus (1962) shows on the particular case of lay-ered media that the effective elastic stiffness tensor c� may carrysome effective anisotropy (also called extrinsic anisotropy) com-puted through a nonlinear process recalled in the next section.

General two-scale homogenization

For the wave propagation problem in general heterogeneous non-periodic media, we will use deterministic homogenization proce-dures as defined in Capdeville et al. (2010b) for the elasticequations and as a special case of Guillot et al. (2010) for the acous-tic equations.For the elastic-wave propagation case, assuming that the density

and elastic properties contain small-scale variations with respect toλmin, the order-zero homogenization approximates the originalequations 1 and 2 with

ρ�u� − ∇ · σ� ¼ f; (13)

σ� ¼ c�∶ϵðu�Þ; (14)

where ρ� is the effective density and c� is the effective fourth-orderelastic stiffness tensor. The effective displacement and stress ðu�; σ�Þapproximate the “real” displacement and stress. Using again a spatiallow-pass filter F λ0ð:Þ with a scale cut-off λ0 (that is, the action ofF λ0ð:Þ removes all variations smaller than λ0; see Appendix B),the effective density is obtained as ρ� ¼ F λ0ðρÞ. The effective elasticstiffness tensor c� cannot be obtained by only filtering the initial elas-tic stiffness tensor c. General deterministic nonperiodic homogeniza-tion (Capdeville et al., 2010a, 2010b) practically computes the

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Figure 4. Experiment 1: Stratified model — subfigures have been grouped in pairs: left, smooth medium and right, rough medium; (a andb) elastic (gray) and acoustic (black) normalized pressure seismograms at receiver 1; (c and d) normalized seismogram gathers for the elasticrun; and (e and f) gathers of the error between elastic and acoustic normalized pressure seismograms.

Table 1. Element sizes and number of GLL points in eachspatial direction for each experiment in rough and smoothcases. For reference, in a homogeneous medium and for nineGLL points per edge of an element, SEM keeps a very goodaccuracy with up to 2λmin per element.

Experiment Element size (m2) GLL points λPminðmÞ

1 (rough) 200 × 25 9 300

1 (smooth) 200 × 200 9 300

2 (rough) 20 × 20 5 200

2 (smooth) 20 × 20 5 200

3 (rough) 10 × 10 5 375

3 (smooth) 50 × 50 9 40

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a) b)

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Figure 5. Experiment 2: Random model — subfigures have been grouped in pairs: left, smooth medium and right, rough medium; (a andb) elastic (gray) and acoustic (black) normalized pressure seismograms at receiver 1; (c and d) normalized seismogram gathers for the elasticrun; and (e and f) gathers of the error between elastic and acoustic normalized pressure seismograms.

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0

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e (s

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Pressure error at depth 100 m

a) b) c) d)

e) f) g) h)

Figure 6. Experiment 3: SEAM-based model — subfigures have been grouped in pairs: left, smooth medium and right, roughmedium; (a and b)and (c and d) elastic (gray) and acoustic (black) normalized pressure seismograms for receivers 1 and 2, respectively; (e and f) seismogram gathersfor the elastic run for reference; and (g and h) seismogram gathers for the error between elastic and acoustic runs.

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effective elastic stiffness tensor through a third-order tensor χ —called the first-order corrector — which is solution to the so-calledstarting cell problem:

∇ · ðHÞ ¼ 0; (15)

H ¼ c∶G; (16)

G ¼ Iþ 1

2ð∇χþ T∇χÞ; (17)

hχi ¼ 0; (18)

with periodic boundary conditions over the “cell,” where h:i is theaveraging operator over the cell (see Appendix B) and where the cellcan be taken as the whole domain. The effective elastic stiffness ten-sor c� is then obtained from the two fourth-order intermediate tensorsG and H with

c� ¼ F λ0ðHÞ∶F λ0ðGÞ−1: (19)

A complete description of deterministic nonperiodic homogenizationfor elastic waves can be found in Capdeville et al. (2010a) (1D case),Capdeville et al. (2010b) (2D P-SV case), and an example of its ac-curacy is given in Appendix C. The effective tensor is in general fullyanisotropic, even for the isotropic fine-scale model, which has beenwell known since the work of Backus (1962).For acoustic media, the same homogenization procedure can be

carried out. The involved theoretical development is mathematicallythe same as the one that can be found in Guillot et al. (2010) forelastic SH-waves in 2D. Applied to the acoustic case, we find that,to compute the effective acoustic wavefield, one has to solve for theeffective potential q� and displacement u�:

1

κ�q� − ∇ · u� ¼ g; (20)

u� ¼ L� · ∇q�; (21)

where 1∕κ� ¼ F λ0ð1∕κÞ is once again the filtered version of 1∕κ,and L� is now a second-order tensor that may carry the effectiveanisotropy. This tensor L� is computed through the acoustic versionof the cell problem (as defined in Guillot et al., 2010), namely, bysolving over the cell for the vector χ

∇ · ðPÞ ¼ 0; (22)

P ¼ 1

ρQ; (23)

Q ¼ Iþ ∇χ; (24)

hχi ¼ 0; (25)

with periodic boundary conditions over the cell, where h:i is theaveraging operator over the cell (see Appendix B), and where once

again the cell is usually taken as the whole domain Ω. The effectivetensor L� is then obtained from the two second-order intermediatetensors Q and P with

L� ¼ F λ0ðPÞ · F λ0ðQÞ−1: (26)

In general, L� is not proportional to the identity, thus leading to ananisotropic effective density. This may appear unusual for the seis-mology community; however, such an anisotropic effective densityhas already been encountered for acoustic waves propagating in theso-called sonic crystals, i.e., in fluid with an embedded periodiclattice of cylindrical scatterers (de Hoop, 1995; Cummer andSchurig, 2007; Torrent and Sánchez-Dehesa, 2008) or in layeredmedia with mixed solids and fluids (e.g., Schoenberg, 1984).At this point, we already see that similar fine-scale structures may

behave very differently at the effective scale level depending on thetype of wave propagation considered (acoustic or elastic). Fromrelations 22–25, it appears that L� depends only on the spatial dis-tribution of the fine-scale density ρðxÞ and not on the bulk modulusdistribution κðxÞ. On the other hand, from equations 16–18, it ap-pears that the effective elastic stiffness tensor c� only depends onthe fine-scale elastic stiffness tensor distribution cðxÞ and not on thefine-scale density distribution ρðxÞ. From this observation, onecould imagine a medium with a homogeneous fine-scale elastic dis-tribution and a heterogeneous density distribution. This would leadto isotropic effective parameters in the elastic case and to largeanisotropic effective parameters in the acoustic case. To the otherextreme, a heterogeneous fine-scale elastic distribution and a homo-geneous density distribution would lead to anisotropic effectiveparameters in the elastic case and to isotropic effective parametersin the acoustic case. Moreover, the fact that the effective “densitytensor” L� is only a second-order tensor in 2D allows only an el-liptic anisotropy, whereas the effective c� is a fourth-order tensor in2D allowing more different types of anisotropy as illustrated in avery simple case in Figure 7b and 7c.As we can see, if the upscaling problems of the elastic- and

acoustic-wave cases are formally very similar, then they involve dif-ferent quantities. From the two upscaling problems, there is no rea-son to expect the same properties from an elastic upscaling and froman acoustic upscaling. Therefore, even if we can build related acous-tic and elastic problems at a small scale, the respective effectiveacoustic and elastic media, the ones indeed seen by the wavefield,have little chance to be related, which explains the numerical obser-vations made in the previous section. To investigate this aspect fur-ther, we study in the next section a simple particular medium case.

Isotropic stratified media

A very useful medium configuration in seismic wave modelingand upscaling is the isotropic stratified medium. For the upscalingpart, this layered medium case reduces to a 1D problem and dis-plays analytical expressions for the effective coefficients as shownby Backus (1962) for the elastic case. Moreover, when layers followa periodic structure, effective parameters remain constant over thewhole domain. This allows us to build simple test cases leading toeasy interpretation of the results. As an example, we will use theisotropic periodic two-layer horizontally stratified medium de-scribed in Figure 7a. In each layer, the density is set using a contrastof 13% around its average value, and Lamé parameters λ and μ areset using contrasts of 50% and 68% around their average value, re-

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spectively (setting approximately κ ¼ λþ 2μ with a contrast of68% around its average value). Each layer is 25 m thick, settingthe medium’s period to 50 m. The source maximum frequencyis set such that the minimum wavelength is λPmin

¼ 400 m, whichis indeed much larger than the period (ε0 ¼ 0.125). Snapshots of theenergy at t ¼ 5 s are shown in Figure 7b for the elastic case and inFigure 7c for the acoustic case, and they illustrate the effectiveanisotropies in each case. In this experiment, three major observa-tions arise:

1) The elastic (diamond shaped wavefront) effective anisotropy ismore complex than acoustic (elliptic) anisotropy.

2) For similar parameters, the acoustic effective anisotropy is weakcompared with the elastic effective anisotropy.

3) The effective elastic P-wave and acoustic-wave velocities areidentical in the vertical direction.

The first point is directly related to the general homogenization pro-cedures for elastic and acoustic waves developed in the previous

section: The different dimensions of the effective elastic and acous-tic tensors explain the different natures of the effective anisotropies.In the following, we focus on the specific analytic results from

the homogenization for isotropic stratified media, which allow us toexplain in detail the last two observations.For 2D elastic periodic stratified media, effective quantities are

computed over the periodic cell as (Backus, 1962)

c�1111 ¼�λþ 2μ −

λ2

λþ 2μ

�þ�

1

λþ 2μ

�−1�

λ

λþ 2μ

�2

;

(27)

c�2222 ¼�

1

λþ 2μ

�−1; (28)

c�1212 ¼�1

μ

�−1; (29)

0

5

10

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th (

km)

0150Offset (km)

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5

10

Dep

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0150Offset (km)

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5

10

Dep

th (

km)

0150Offset (km)

Acoustic t=5.0 s

a)

b) c) d)

Figure 7. Visualization of effective anisotropy on a two-layer periodic medium in the elastic and acoustic cases. (a) Original elastic medium,(b) snapshot of the elastic wavefield, (c) snapshot of the acoustic wavefield, and (d) snapshot of the acoustic wavefield computed in a mediumdevised from the original elastic medium with the intent to respect effective kinematics in horizontal and vertical directions.

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c�1122 ¼�

1

λþ 2μ

�−1�

λ

λþ 2μ

�; (30)

ρ� ¼ hρi; (31)

where h:i denotes the averaging operator over the periodic cell (seeAppendix B).In 2D acoustic periodic stratified media, the effective density ten-

sor’s analytical expression is computed similarly as for a static sca-lar problem (Lurie and Cherkaev, 1986; Bosse and Showalter, 1989;Hornung, 1992), and the effective parameters are then given by

L� ¼� h1ρi 0

0 h1ρi�; (32)

1

κ�¼

�1

κ

�: (33)

Therefore, to upscale such a medium, only three simple quantitiesneed to be computed, which are hρi, h1∕ρi, and h1∕κi.At this point, the effective kinematic equivalence between elastic

P-waves and acoustic waves in the vertical direction (observation 3)can easily be confirmed. Indeed, noticing that for our acoustic pa-rameterization κ ¼ ρV2 ¼ ρVP

2 ¼ λþ 2μ and using equations 28,31, L�

22-expression from equations 32 and 33, we obtain

VPV� ¼

ffiffiffiffiffiffiffiffiffiffic�2222ρ�

s¼ ffiffiffiffiffiffiffiffiffiffiffi

κ�L�22

p ¼ VV�; (34)

which means that elastic P-waves and acoustic waves travel verti-cally with the same effective velocity (respectively, VPV

� and VV�).

To explain the differences in amplitude of effective anisotropies(observation 2), we use a “Thomsen-like” (Thomsen, 1986) meas-urement of anisotropy ϵ, in the elastic and acoustic cases. In theelastic case, ϵ is exactly the Thomsen parameter ϵ ¼ ðc�1111−c�2222Þ∕ð2c�2222Þ; which, for a two-layered medium, can be computedas

ϵ ¼ ν2κ2ð1 − ν2κÞ

�1 −

�hλihκi

νλνκ

�2�; (35)

where κ ¼ λþ 2μ with λ and μ the Lamé parameters; hλi and hκidenote the average values of λ and κ, respectively; and νλ and νκ aretheir respective contrasts.In the acoustic case, we similarly use ϵ ¼ ðL�

11 − L�22Þ∕ð2L�

22Þ;which, for a two-layered medium, can be computed as

ϵ ¼ ν2ρ2ð1 − ν2ρÞ

; (36)

where νρ is the contrast on the density.First, we note that relations 35 and 36 show some formal similar-

ities because both of them relate the effective anisotropy to contrastson the heterogeneities with the following pattern: The higher the con-trasts are (ν → 1), the more important the effective anisotropy is. Inour example, the contrast on the density was much smaller than con-trasts on the Lamé parameters, and we therefore have ϵ ≃ 38.7% in

the elastic case corresponding to a strong anisotropy, whereas wehave ϵ ≃ 0.9% in the acoustic case, corresponding to almost noanisotropy as observed. However, equations 35 and 36 show directlythat the important quantities related to effective anisotropy are fun-damentally different between the elastic and the acoustic cases. In theelastic case, effective anisotropy depends only on contrasts of theLamé parameters, whereas in the acoustic case, effective anisotropydepends only on contrasts of the density. Therefore, if effectiveanisotropies were already of different kinds (observation 1), relations35 and 36 show very clearly in the periodic two-layer media case thatthey are also different in their strength as density’s contrasts andLamé parameters’ contrasts are not necessarily correlated, especiallyfor shallow geologic structures of the earth.Figure 7d shows the result for an attempt to build a two-layer

acoustic medium at the microscopic scale with effective kinematicequivalence, at least for the radial and transverse directions with theelastic case (i.e., to obtain identical values of ϵ for the elastic andacoustic cases). To obtain such a selective effective kinematic equiv-alence (the acoustic effective anisotropy remains elliptic, which stilldoes not account correctly for the diamond-shaped elastic effectiveanisotropy), the construction at the microscopic level of acousticparameters from the elastic parameters can be easily devised (in anonuniqueway though, and using the effective elastic parameters) fortwo-layer periodic media. However, such a mapping between theelastic and acoustic parameters is only a mathematical transforma-tion, and the physical meaning of the acoustic parameters is lost here.Unfortunately, only a simple kinematic equivalence can be achievedfollowing this process. Moreover, this mapping in a two-layer casecannot be easily extended to more complex cases, such that we didnot search any further for such kinematic equivalences.At last, very similar results can be obtained for the continuous

case of experiment 1. In particular, the vertical effective kinematicequivalence can be seen through the dependence of the error withoffset: For low offsets, waves propagate nearly vertically and nodelay has to be observed, whereas for a larger offset, full anisotropyhas to be considered for phase comparison of the signals. Then, thelower contrast on the density compared with contrasts on the elasticmoduli implies that acoustic effective anisotropy is weaker thanelastic effective anisotropy, which corresponds to the observed de-lay of the acoustic signal toward the elastic signal.

CONCLUSIONS

The acoustic approximation of elastic waves is a common approxi-mation in exploration geophysics. In the present work, we haveaddressed the validity of this approximation regarding small hetero-geneity scales with respect to the wavelength of the wavefield. Theapproximation remains valid for isotropic homogeneous media orisotropic heterogeneous media with smooth variations with respectto the wavefield’s wavelength. However, for heterogeneous roughmedia, i.e., when scales for the variations in the elastic parameters aresmall compared with the wavelengths of the wavefield, this approxi-mation is not reliable anymore. Indeed, forgetting about amplitudeerrors due to unmodeled P-to-S conversions by acoustic equations,phase errors still arise due to differences in kinematics perceivedby the waves between the elastic and acoustic case. Homogenizationprocedures — for acoustic waves on the one hand and for elasticwaves on the other hand — offer a way to understand the differencesin effective kinematics through the computation of effective param-eters and effective anisotropy. Two main reasons arise to explain such

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effective kinematic differences. First, acoustic effective anisotropy,which is carried by an effective anisotropic density, is systematicallyreduced to an elliptic anisotropy, whereas elastic effective anisotropydoes not suffer such restriction. Second, even if acoustic parametersare related to elastic parameters at the original small scale, it is gen-erally not the case anymore for effective parameters because relevantquantities for computing effective parameters (density for the acous-tic case and elastic moduli for the elastic case) are not the same. Ifthese two results are available for general heterogeneous media, theparticular case of stratified periodic media allows for a more detailedinvestigation by offering clear relations between parameters of themicroscale medium and effective anisotropy.Our work shows that if using the acoustic approximation for

numerical experiments for full-waveform inversion is still to be con-sidered possible and pertinent, some care has to be given on whatparameters to work on to fully mimic the elastic case. Indeed,inverting mainly for P-wave parameters (such as VP, eventuallywith anisotropy parameters) is a common strategy. Such a choicein the acoustic case is equivalent to invert only for the density in theelastic case, which is possible but missing a lot of the realistic com-plexity. Inverting for the density, or better for L, is a better choice,which makes it possible to account for some of the elastic complex-ity in the acoustic framework.In the end, using equations of acoustics as an acoustic approxi-

mation for elastic P-waves seemed natural at first, but this approxi-mation remained too simple to keep a good accuracy and a physicalequivalence in highly heterogeneous — or rough — media.

ACKNOWLEDGMENTS

This work was funded by the "Mémé" grant (ANR-10-BLAN-613 MEME) from the French National Research Agency (ANR).Computations were performed on the computer cluster "Erdre"from the Centre de Calcul Intensif des Pays de la Loire (CCIPL).We thank G. Festa for letting us use and modify his 2D spectralelement program and J. Rickett for letting us use and modify the 2Dcrosscut of SEG’s SEAM model. Last but not least, we want tothank Fichtner, Oprsal, and five other anonymous reviewers fortheir numerous constructive comments, which allowed for signifi-cant improvement of the quality of this paper.

APPENDIX A

FROM ELASTIC TO ACOUSTIC EQUATIONS: THEHOMOGENEOUS ISOTROPIC CASE

The only case for which the elastic- and acoustic-wave equationscan exactly be related is for an infinite isotropic homogeneous do-main and for an explosive isotropic source. In such a case, acousticequations derive from elastic equations as follows: In an isotropichomogeneous domain, the elastic medium is fully described by itsdensity ρ and the two Lamé parameters λ and μ, which are constantover the domain. Then, using equation 4 in equation 1, we obtain

ρu − λ∇ · ½ð∇ · uÞI� − μ∇ · ð∇uÞ − μ∇ · T∇u ¼ f ; (A-1)

where f ¼ ∇ðδðx − x0ÞRickerðtÞÞ is the explosive source term, withx0 being the position of the source, δðxÞ is the Dirac distribution,and Ricker ðtÞ is the source time function as a Ricker. Using thefollowing identities:

∇ · ½ð∇ · uÞI� ¼ ∇ð∇ · uÞ ; (A-2)

∇ · T∇u ¼ ∇ð∇ · uÞ; (A-3)

∇ · ð∇uÞ ¼ ∇ð∇ · uÞ − ∇ × ð∇ × uÞ ; (A-4)

the elastic-wave equation can be rewritten as

ρu − ∇½ðλþ 2μÞ∇ · u� þ μ∇ × ð∇ × uÞ ¼ f ; (A-5)

where ∇ × u denotes the curl operator over vector u.Assuming that the source is not generating S-waves, only P-

waves are propagating in Ω, and knowing that P waves are irrota-tional, we have∇ × u ¼ 0. In such a case, the displacement vector uor the velocity vector v ¼ u derives from a potential q. The densitybeing constant, we can choose the potential q such as

v ¼ u ¼ 1

ρ∇q: (A-6)

Differentiating equation A-5 over time, using equation A-6, assum-ing the source also derives from a potential g such that

f ¼ ∇ððλþ 2μÞgÞ ; (A-7)

which is possible with the explosive source mentioned above, andsetting κ ¼ λþ 2μ ¼ ρVP

2, we obtain

∇�1

κq − ∇ · v − g

�¼ 0 ; (A-8)

which implies equation 7. Therefore, in an infinite homogeneousisotropic medium with the acoustic density set to the elastic density,the sound speed set exactly to the elastic P-wave velocity V ¼ VP,and assuming that no S-wave has been generated by the source,acoustic equations 7 and 8 have the same solution as the elastic-wave equations.

APPENDIX B

FILTERING AND AVERAGING OPERATORS

We define here the averaging h:i and filtering F λc ð:Þ operators.For a quantity hðxÞ defined over a 2D cell Ω, the correspondingaverage is defined as

hhi ¼ 1

jΩjZΩhðxÞdx; (B-1)

where jΩj is a measure (e.g., the area) of the cell.For any function h, we define its 2D Fourier transform as

hðkÞ ¼ZR2

hðxÞ expðik · xÞdx ; (B-2)

where x is the position vector, and k is the wavenumber vector. Forany wavenumber vector k, we define its associated wavelengthλk ¼ 2π∕jkj. To remove any spatial variations smaller than λc ofa function hðxÞ, we define the low-pass-filter operator F λcð:Þ as

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F λcðhÞðxÞ ¼ZR2

hðx 0Þwλcðx − x 0Þdx 0; (B-3)

where wλc is a wavelet. Ideally, wλc is such that

wλcðkÞ ¼1 for jkj ≤ kc;0 for jkj > kc;

(B-4)

where kc ¼ 2π∕λc. In practice, we use a wavelet with a finite spatialsupport. We first define the mother wavelet wðxÞ, such that itspower spectrum is

wðkÞ ¼

8><>:

1 for jkj ≤ a;12

h1þ cos

π jkj−a

b−a

�i;

0 for jkj ∈ ða; bÞ;(B-5)

where a and b are two real numbers of approximately 1 defining thetapper transition between 1 and 0 of the low-pass filter. The waveletin the space domain is then obtained with a Hankel transform:

wðxÞ ¼Z þ∞

0

wðkÞJ0ðjkjjxjÞjkj djkj; (B-6)

where J0 is the Bessel function of the first kind of order zero. Last,we define the filter wavelet wλc ðxÞ of corner frequency kc ¼ 2π∕λcas wλcðxÞ ¼ kcwðkcxÞ.

APPENDIX C

NUMERICAL PRECISION AND PHASE SHIFT

We show here that phase shifts observed between elastic- andacoustic-pressure signals in the rough cases are not numerical arti-facts due to SEM simulations and are related to small-scale hetero-geneities through the homogenization procedure. To show that thereis no precision loss due to the spectral element procedure, we useexperiment 2 (random medium) in the rough case configurationwith two different numerical setups. The first setup is defined aspreviously, with 800 × 900 square elements of 20 × 20 m2 withconstant elastic properties and with five GLL points per space di-rection. The second setup is different from the first only by refiningthe mesh (and time step) by a factor of two in each direction. Fig-ure C-1a shows that in the first setup, the reflected wave recorded atreceiver 1 had already numerically converged and the relative errorjðp1 − p2Þ∕p2j on the whole pressure signal is no more than 1% forelastic and acoustic runs.To numerically show that homogenization theory fully explains

the observed phase shifts between elastic and acoustic waveforms,we homogenize the random medium of experiment 2 in elastic andacoustic cases, with a homogenization parameter of ε0 ¼ 0.5

(which is generally small enough to obtain good accuracy). Then,we compute the wavefield in the elastic and acoustic cases using thehomogenized media and with the same numerical setup as in thepaper’s body (the first setup). We then compare pressure signalscomputed in the original medium and in the homogenized mediumfor the elastic and acoustic cases. Figure C-1b shows the waveformrecorded at receiver 1 in the time window corresponding to the re-flected wave. Further convergence analysis of the homogenizationprocedure can be found in Capdeville et al. (2010b) for the elasticcase and Guillot et al. (2010) for the SH-elastic or acoustic cases.

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Figure C-1. Experiment 2, rough medium: Numerical convergenceof the SEM computations (a) and of the homogenization procedure(b) on the reflected wave recorded at receiver 1.

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